Twelve-monthly values of rainfall since 1883 at Manilla NSW yield the four moments of their frequency distributions: mean, variance, skewness, and kurtosis. I plotted the history of each moment (when smoothed) in **an earlier post**.

Here, I compare the moments in pairs. Connected scatterplots reveal the trajectory of each relationship with time.

**Some linear and cyclic trends persist through decades, but none persists through the whole record.**

The first image is an index to the suite of six graphs of pair-wise relationships that I present below.

# Tag Archives: kurtosis

# Rainfall kurtosis vs. HadCRUT4, revised

## The **kurtosis** of annual rainfall at Manilla NSW forms a time-series that matches the time-series of global surface temperature when detrended.

**[REVISED:**

**Earlier posts were based on rainfall data sets that were too small. Estimates of kurtosis and skewness were unstable. ****For details please read “Rainfall kurtosis matches HadCRUT4” and “Rainfall kurtosis vs. HadCRUT4: scatterplots”.]**

## The variables

These two climate variables have little in common. Manilla, NSW, is a single station that has a 134-year record of daily rainfall only. That yields estimates of rainfall kurtosis, an indicator of the relative frequency of extreme values.

HadCRUT4 is one of several century-long estimates of near-surface temperature for the whole world. [See Note below: “Data Sources”.]

## The visual match of the patterns

The first graph (a dual-axis line chart) shows that these two variables have similar patterns of variation over time.

I found the best visual match by:

* scaling 0.5 units of Manilla rainfall kurtosis to 0.1° of detrended HadCRUT4 temperature;

* aligning the kurtosis value of -0.3 units with the zero of detrended temperature;

* lagging the rainfall by two years.

Features that the two patterns have in common are:

* matching main peaks at 1897, 1942 and 2005, each higher than the one before;

* persistent low values in the 1910’s, 1920’s, 1950’s, 1960’s, 1970’s and early 1980’s;

*some matching minor peaks and troughs.

## The correlation chart

The second graph is a correlation chart. The linear regression of kurtosis on detrended temperature has the reasonable R-squared value of 0.67.

As I have made it a connected scatterplot, you can see how the relation has changed through time. From the first data point in 1898 (in **red**) both variables decreased together to the lowest temperature in 1910. Both peaked in 1942, having risen since 1920, later falling until 1955-56. The final rise to the highest peak (2005) was continuous from 1984 for temperature, but the rise in kurtosis was not. It fell slightly in 1990, then remained static until 1998.

All rainfall figures actually came two years earlier. **[See note below: “Manilla’s 2-year lead”.]** The assigned two-year lag not only makes peaks match on the first graph. It sharpens the reversals on the second graph. On a trial connected scatterplot without lag, these reversals had been smooth clockwise curves.

## What it means

### As evidence of extreme behaviour in climate

It is said that more extremes in climate will occur as the world becomes warmer. The evidence is not strong. Most data sets are overwhelmed by noise, and “extreme” is seldom defined with rigor.

In the present case, I believe that the definition of “extreme” that I use is sound: that is, the kurtosis of a frequency-distribution. The instability of kurtosis when based on my small samples had been an issue. In this revision I have increased the sample population size from 21 to 125.

My rainfall data set that displays more and less extreme behaviour is not general but local. It can merely suggest that data elsewhere may reveal functional relationships.

### De-trended global temperature

# Moments of Manilla’s 12-monthly Rainfall

**REVISED, WITH MORE PRECISE DATA**

**Supersedes the post “Moments of Manilla’s Annual Rainfall Frequency” ****(15 November 2017). ****This post includes twelve times as much data.[See Note below: “Data handling”]**

## Comparing all four moments of the frequency-distributions

Yearly rainfall for Manilla, NSW, has varied widely from decade to decade, but it is not only the mean amounts that have varied. Three other measures have varied, all in different ways.

I based the graph on 125-month (decadal) sub-populations of the 134-year record. I plotted data for every month, at the middle month of each sub-population.

I analysed each sub-population as a **frequency-distribution**, to give values of the four **moments**: mean (drawn in **indigo**), variance (drawn in **orange**), skewness (drawn in **green**) and kurtosis (drawn in **blue**).

[For more about the moments of frequency-distributions, see the post: **“Kurtosis, Fat Tails, and Extremes”**.]

Each trace of a moment measure seems to have a pattern: they are not like random “noise”. Yet each trace is quite unlike the others.

Twenty-first century values are on the right. They are remarkable in three of the four moments. First, the mean rainfall (indigo) stays near the long-term mean, which has seldom happened before. By contrast, two moments are now near historical extremes: variance (orange) is very low and kurtosis (blue) very positive. Skewness (green) is rather negative.

To my knowledge, such a result has not been observed or predicted, or even suspected, anywhere.

[**Note.** The main difference from the earlier 4-moment graph based on more sparse data is that skewness does not trend downward.]

### The mean 12-monthly rainfall (the first moment)

The first moment of the frequency-distribution of 12-monthly rainfall is the mean, or average. It measures of the **amount** of rain.

As I have **shown before**, the rainfall was low in the first half of the 20th century, and high in the 1890’s, 1950’s and 1970’s. Rainfall crashed in 1900 and again in 1980.

## 12-monthly rainfall variance (the second moment)

# Annual Rainfall Extremes at Manilla NSW: V

## V. Extremes marked by high kurtosis

This graph shows how the extreme values of annual rainfall at Manilla, NSW have varied, becoming rarer or more frequent with passing time.

The graph quantifies the occurrence of extreme values by the **kurtosis** of 21-year samples centred on successive years.

The main features of the pattern are:

* Two highly leptokurtic peaks, showing times with strong extremes in annual rainfall values. One is very early (1897) and one very late (1998).

* One broad mesokurtic peak, in 1938, showing a time with somewhat weaker extremes.

* Broad platykurtic troughs through the 1910’s, 1920’s, 1950’s, 1960’s and 1970’s, decades in which extremes were rare.

All these features were evident in the cruder attempts to recognise times of more and less occurrence of extremes in Parts **I**, **II**, **III** and **IV** of this series of posts. This graph is more precise, both in quantity and in timing.

**Superseded**

**ALL the results shown in this post are based on sparse data. They are superseded by results based on much more detailed data in the post** **“Relations Among Rainfall Moments”**.

However, kurtosis (the fourth moment of the distribution) does not distinguish extremes above normal from those below normal. It is known that some early dates at Manilla had extremes that were above normal, and some late dates had extremes that were below normal.

### Use of skewness

Extremes above normal are distinguished from those below normal by the third moment of the distribution, that is, the **skewness**.

The post **“Moments of Manilla’s Yearly Rainfall History”** shows graphs of the time sequence of each of the four moments, including the skewness (copied here) and the kurtosis ( the main graph, copied above). The skewness function, like the kurtosis function, relates to the most extreme values of the frequency distribution, but to a lesser extent (by the third power, not the fourth).

I have shown the combined effect of kurtosis and skewness on the occurrence of positive and negative extremes in this data set in the connected scatterplot below.

The early and late times of strong extremes were times of strongly positive and strongly negative skewness respectively. As kurtosis fell rapidly from the initial peak (+0.9) in 1897 to slightly platykurtic (-0.4) in 1902, the skewness also fell rapidly, from +0.7 to +0.3.

Much later, in mirror image, values were almost the same in 1983 as in 1902, then kurtosis rapidly rose while skewness rapidly fell, until kurtosis reached +0.9 and skewness -0.3 by 1998.

Between 1902 and 1983, while kurtosis remained below -0.2, the pattern was complex. In the decades of strong platykurtosis (below -0.9) there were extremes of skewness: +0.7 in 1919 and -0.3 in 1968.

Note that the skewness range was as high in times of low kurtosis as in times of high kurtosis, and the same applies to kurtosis range in relation to skewness. Conversely, when either moment was near its mean, the range of the other was not high.

**See also:**

**“Rainfall kurtosis matches HadCRUT4”** and **“Rainfall kurtosis vs. HadCRUT4 Scatterplots”**.

# Rainfall kurtosis vs. HadCRUT4 Scatterplots

## These scatterplots and Connected Scatterplots support a relationship between the kurtosis of annual rainfall at Manilla NSW and the de-trended smoothed HadCRUT4 series of global temperatures.

[**SUPERSEDED**

This post had inadequated data. It is now superseded by a section in the post **“Rainfall kurtosis vs. HadCRUT4, revised”** of 20 May 2018.]

## The raw data, as observed

The first scatterplot compares (y-axis) all the calculated unsmoothed values of kurtosis of annual rainfall at Manilla, NSW with (x-axis) the unsmoothed values of the HadCRUT4 series of global near-surface temperature at those dates.

[I have plotted rainfall values lagged by five years on all of the scatterplots shown. This lagging makes little difference to the first two scatterplots.]

On this first graph, the fitted linear trend barely supports a positive relation of kurtosis to temperature. The slope is low (1.05) and the R-squared only 0.16. There is an aberrant cloud of points in the top left corner.

## The raw data, HadCRUT4 de-trended

This graph takes a first step towards a better model for the relationship: the secular trend of the temperature series (that is, the global warming) is removed. For comparison, I have not re-scaled the x-axis.

Although still very weak, the relation is much enhanced. The slope (2.35) is twice as steep and the R-squared (0.24) increased by 50%.

## Smoothed data, HadCRUT4 de-trended

This third graph uses smoothed data. The HadCRUT4 series is “decadally-smoothed” (as published) with a 21-point binomial filter to remove high frequency noise. The rainfall data, already damped by its 21-year sampling window, has been further smoothed with a 9-point Gaussian filter.

This graph is a **Connected Scatterplot**, that shows the trajectory of the rainfall-temperature relation with the passing of time.

Smoothing both data sets has given a much closer relation. The R-squared value is almost doubled again, to 0.43, and the slope is increased to 3.70. The date labels show that the relation before 1910 was different from that at later dates. (This had also been clear in the Dual axis line chart, copied here, from the post **“Rainfall Kurtosis Matches HadCRUT4”**.)

## Smoothed data, HadCRUT4 de-trended, from 1908 to 2002

In this final graph, I have discarded the first eleven years. The linear regression based on smoothed values from 1908 to 2002 has a steep slope of 5.21 and a respectable R-squared value of 0.84.

I had prepared similar graphs for lag values of rainfall kurtosis from zero up to nine. The lag value of five years tends to maximise the slope and the R-squared values.

Choice of a five-year lag tends to form hair-pin loops in the trace, while shorter lags give wider clockwise loops and longer lags give wider anti-clockwise loops.

The lag value of five years implies that the Manilla annual rainfall kurtosis value for a given year matches the de-trended HadCRUT value that occurs five years later.

[Back to the main post on this topic: **“Rainfall kurtosis matches HadCRUT4”**.]